math 1b final

sin(0)

0

cos(0)

1

tan(0)

0

sin(π/6)

1/2

cos(π/6)

√3/2

tan(π/6)

√3/3

sin(π/4)

√2/2

cos(π/4)

√2/2

tan(π/4)

1

sin(π/3)

√3/2

cos(π/3)

1/2

tan(π/3)

√3

sin(π/2)

1

cos(π/2)0

tan(π/2)

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tests

1. p-series

2. geometric series

3. alternating series (AST)

4. n^th term

5. ratio

6. partial sums

7. direct (DCT)

8. intergral (ICT)

9. asymptotic

tests based on series

1. p-series

2. geometric series

3. alternating series (AST)

tests based on sums

1. n^th term

2. ratio

3. partial sums

comparison tests

1. direct (DCT)

2. intergral (ICT)

3. asymptotic (ACT)

p-series test

geometric series test

alternating series test (AST)

n^th term test

ratio test

direct comparison test (DCT)

integral comparison test (ICT)

asymptotic comparison test (ACT)

Taylor Series:

ln(1+x)

Taylor Series:

1/(1-x)

Taylor Series:

e^x

Taylor Series:

cos(x)

Taylor Series:

sin(x)

Taylor polynomial formula

a sub k formula

a sub n coefficients obtained by:

harmonic series formula

error bound for converging alternate series

power series formula / “infinite degree polynomial”

n^th degree Taylor polynomial

partial sums test

disk volume formula

Volume = π r² Δx

washer volume formula

Volume = (π R² - π r²) Δx

cylindrical shell volume formula

2π r h Δx

disk method traits

• solid (not hollow)

• dx for HORIZONTAL axis of rotation

• dy for VERTICAL axis of rotation

washer method traits

• hollow

• dx for HORIZONTAL axis of rotation

• dy for VERTICAL axis of rotation

cylindrical shell method traits

• either solid or hollow

• dx for VERTICAL axis of rotation

• dy for HORIZONTAL axis of rotation

probability definition

probability density function

probability that x lies in an interval [a, b]

median value of x

number M such that

mean value of x or the expected value of x

Comparison Theorem

0 ≤ f (x) ≤ g(x)

Comparison Theorem warnings

1/x converge or diverve?

diverge

1/x² converge or diverge?

converge

1/x³ converge or diverge?

converge (check)

trapezoidal sum formula

left- and right-hand sum formulas

Error in Numerical Approximations (trapezoid & midpoint)

mid point sum formula

Error Bounds in Numerical Approximations (left and right hand sums)

work formula

work = mgh

limit definition of an integral

∆x

x_k

p-integral rule

if p = 1

function diverges

product rule

triangle inequality

circle formula

add later: common derivatives, similar triangles, improper integrals, & steps for common problems

characteristic equation of second-order differential equatio

if the characteristic equation has two distinct real roots r1 and r2, then the general solution is

If the characteristic equation has one repeated real root r1, then the general solution is

If the characteristic equation has two complex roots α ± βi, then the general solution is

Euler’s formula

mass-spring system conditions

b≥0 (b = 0 means no friction)

c>0